MinT - Best Known (21, 54, s)-Nets in Base 16

Best Known (21, 54, s)-Nets in Base 16

(21, 54, 71)-Net over F₁₆ — Constructive and digital

Digital (21, 54, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (3, 36, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(21, 54, 104)-Net in Base 16 — Constructive

(21, 54, 104)-net in base 16, using

6 times m-reduction [i] based on (21, 60, 104)-net in base 16, using
- base change [i] based on digital (9, 48, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(21, 54, 129)-Net over F₁₆ — Digital

Digital (21, 54, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 54, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(21, 54, 4408)-Net in Base 16 — Upper bound on s

There is no (21, 54, 4409)-net in base 16, because

1 times m-reduction [i] would yield (21, 53, 4409)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6603 338026 466811 898242 144456 687425 504939 504320 111751 744924 123161 > 16⁵³ [i]